A method for the numerical solution of parabolic equations with nonlinear diffusion

We propose a finite difference scheme for the diffusion equation, ( *) ut = d(u)Δu + f(μ), on a general spatial domain of Rm, m ≥ ≥ 1, d(u) is a bounded positive smooth function. For the numerical solution of (*) one usually uses a finite difference method based on the well-known θ-method, which requires a factorization of a matrix at each time step. Here we propose a numerical scheme in which we need a single factorization of a matrix for each time level. We prove that if W is an invariant region for (*), it is also invariant for the proposed method. Comparisons between our scheme and the explicit/implicit Euler method are made. We give an error bound which implies the first order convergence of the method and shows that the error does not exceed diam (W) for t → +∞. Finally, we show a numerical application